The problem that comes from each character only being able to heal themselves comes from a little thing called tempo. (If you play a lot of MtG, you already know what I'm talking about.) If you take a turn to heal, and the enemy takes a turn to attack, then you have gotten nowhere and you've either lost HP, MP or both. The enemy, on the other hand, has lost nothing. If you take a turn to attack, then you hurt the enemy, but you leave yourself open. If you lose (proportionally) more HP than your enemy, you have also gained nothing. The only way to win is to have strong heals, things that heal around 75% of your HP, weak enemy attacks that take away less than 50% of your HP, and alternate from healing and attacking every turn. Here's how it would shake out:
T1: HP: 100. Enemy HP (EHP): 200. You attack, enemy attacks.
T2: HP: 60. EHP: 150. You heal, enemy attacks.
T3: HP: 60. EHP: 150.You attack, enemy attacks.
T4: HP: 10. EHP: 100. You heal, enemy attacks.
T5: HP: 45. EHP: 100. You attack, enemy attacks.
T6: HP: 5. EHP: 50. You attack, enemy is defeated.
Once you figure out the rhythm, every battle becomes boring. Attack until you need to heal, then heal until you've regained enough HP to attack again. Rinse and repeat.
As for a solution, you could have players go twice. That way they have roughly three options a turn: all out attack, attack and heal, or all out heal. Battles would typically go one of two ways: all out attack one turn followed by an all out heal the next, or an attack and heal every turn.
If you're trying to implement this into RPG Maker, just give each party member an turn and make most skills inflict something akin to a "charging" status.
For example:
On Turn 1:
Party Member 1 does a minor heal and a minor attack.
Party Member 2 does a major heal.
Party Member 3 does a major attack.
Party Member 4 does a minor heal and a major attack.
On Turn 2:
Party Member 1 does a minor heal and a minor attack.
Party Member 2 does a major attack.
Party Member 3 does a major heal.
Party member 4 recovers from doing too many things in one turn, and only gets a minor heal.
And so on.
This way, you can utilize half-turns for healing while still keeping up the assault. It's still probably a good idea to have regen and absorb moves, but those can be implemented into this system. You can still have it so each player can only heal themselves. (It probably wouldn't be a bad idea to give enemies 2 turns, as well...)
Good luck with your game, and remember that balance and fun are key.